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Topological introduction to nonlinear analysis / Robert F. Brown.

By: Publication details: Switzerland : Springer, 2014.Edition: 3rd edDescription: x, 240 p. ; illustrationsISBN:
  • 9783319117935
Subject(s): DDC classification:
  • 515.7 23 B879
Contents:
Part I Fixed Point Existence Theory -- 1. The Topological Point of View -- 2. Ascoli-Arzela Theory -- 3. Brouwer Fixed Point Theory -- 4. Schauder Fixed Point Theory -- 5. The Forced Pendulum -- 6. Equilibrium Heat Distribution -- 7. Generalized Bernstain Theory -- Part II Degree Theory -- 8. Brouwer Degree -- 9. Properties of the Brouwer Degree -- 10. Leray-Schauder Degree -- 11. Properties of the Leray-Schauder Degree -- 12. The Mawhin Operator -- 13. The Pendulum Swings back -- Part III Fixed Point Index Theory -- 14. A Retraction Theorum -- 15. The Fixed Point Index -- 16. The Tubulur Reactor -- 17. Fixed Points in a Cone -- 18. Eigenvalues and Eigenvectors -- Part IV Bifurcation Theory -- 19. A Separation Theorem -- 20. Compact Linear Operators -- 21. The Degree Calculation -- 22. The Krasnoselskii-Rabinowitz Theorem -- 23. Nonlinear Strum Liouville Theory -- 24. More Strum Liouville Theory -- 25. Euler Buckling -- A. Singular homology -- B. Additivity and product properties -- C. Bounded linear operators -- D. Metric implies paracompact -- References -- Table of symbols -- Index.
Summary: This third edition of A Topological Introduction to Nonlinear Analysis is addressed to the mathematician or graduate student of mathematics - or even the well-prepared undergraduate - who would like, with a minimum of background and preparation, to understand some of the beautiful results at the heart of nonlinear analysis. Based on carefully-expounded ideas from several branches of topology, and illustrated by a wealth of figures that attest to the geometric nature of the exposition, the book will be of immense help in providing its readers with an understanding of the mathematics of the nonlinear phenomena that characterize our real world. For this third edition, several new chapters present the fixed point index and its applications. The exposition and mathematical content is improved throughout. This book is ideal for self-study for mathematicians and students interested in such areas of geometric and algebraic topology, functional analysis, differential equations, and applied mathematics. It is a sharply focused and highly readable view of nonlinear analysis by a practicing topologist who has seen a clear path to understanding.
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Includes bibliographical references and index.

Part I Fixed Point Existence Theory --
1. The Topological Point of View --
2. Ascoli-Arzela Theory --
3. Brouwer Fixed Point Theory --
4. Schauder Fixed Point Theory --
5. The Forced Pendulum --
6. Equilibrium Heat Distribution --
7. Generalized Bernstain Theory --

Part II Degree Theory --
8. Brouwer Degree --
9. Properties of the Brouwer Degree --
10. Leray-Schauder Degree --
11. Properties of the Leray-Schauder Degree --
12. The Mawhin Operator --
13. The Pendulum Swings back --

Part III Fixed Point Index Theory --
14. A Retraction Theorum --
15. The Fixed Point Index --
16. The Tubulur Reactor --
17. Fixed Points in a Cone --
18. Eigenvalues and Eigenvectors --

Part IV Bifurcation Theory --
19. A Separation Theorem --
20. Compact Linear Operators --
21. The Degree Calculation --
22. The Krasnoselskii-Rabinowitz Theorem --
23. Nonlinear Strum Liouville Theory --
24. More Strum Liouville Theory --
25. Euler Buckling --
A. Singular homology --
B. Additivity and product properties --
C. Bounded linear operators --
D. Metric implies paracompact --
References --
Table of symbols --
Index.

This third edition of A Topological Introduction to Nonlinear Analysis is addressed to the mathematician or graduate student of mathematics - or even the well-prepared undergraduate - who would like, with a minimum of background and preparation, to understand some of the beautiful results at the heart of nonlinear analysis. Based on carefully-expounded ideas from several branches of topology, and illustrated by a wealth of figures that attest to the geometric nature of the exposition, the book will be of immense help in providing its readers with an understanding of the mathematics of the nonlinear phenomena that characterize our real world. For this third edition, several new chapters present the fixed point index and its applications. The exposition and mathematical content is improved throughout. This book is ideal for self-study for mathematicians and students interested in such areas of geometric and algebraic topology, functional analysis, differential equations, and applied mathematics. It is a sharply focused and highly readable view of nonlinear analysis by a practicing topologist who has seen a clear path to understanding.

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